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Graphing a Quadratic

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Warm-up

- Determine the following algebraically (no calculator)
- vertex
- x- and y-intercepts.
- Is the vertex a max or min? How would you know without graphing?
- Identify intervals of increasing/decreasing.

- Write f in standard form (complete the square)

Section 3.2 Polynomial Functions and Models

- Polynomial Functions and their Degree
- Properties of Power Functions
- Zeros of a Polynomial and Multiplicity
- Building Polynomials
- End Behavior of Polynomials
- Leading Coefficient Test

- Analyzing the Graph of a Polynomial Function

A polynomial function is a function of the form

where the coefficients are real numbers

and n is a nonnegative integer.

The degree of the function is the largest power of x.

Factored form: Add the degrees

Determine which of the following are polynomial functions. For those that are, state the degree.

1.

2.

3.

4.

5.

Determine which of the following are polynomial functions. For those that are, state the degree.

6.

7.

8.

9.

10.

Determine which of the following are polynomial functions. For those that are, state the degree.

T; 56. T; 3

2. F7. T; 7

3. T; 58. T; 3

4. F9. F

5. T; 010. F

A power function is of the form

If n is odd integer

If n is even integer

Symmetry:

Domain:

Range:

Key Points:

- Factor TheoremIf is a polynomial function, the following are equivalent statements:
- r is a real number for which:
- r is called a zero or root of
- r is an x-intercept of the graph of
- is a factor of

Example: Factor

For each factor, summarize properties 1-4 of the Factor theorem..

Definition: The multiplicity of a zero is the degree of the factor

Notation:

If f has a factor we say is a zero of multiplicity

Example: Identify the zeros and their multiplicities of:

Degree of f =

Graph f(x).

Example – continued.

What does the value of m tell us about the graph ?

Analyze the graph of

What happens at the zero as m gets large?

Use your graphing calculator to graph the following:

1)

2)

The graph flattens out at the zero as the multiplicity increases.

State the degree of this polynomial.

How many zeros does this function have?

Given that f has zeros with multiplicity

we can write:

Write a polynomial with these properties:

1) Degree 4: Zeroes at: -5, -4, 0, 2, (multiplicity 1)

2) Degree 3: Zeroes at: -3, multiplicity 2;

5, multiplicity 1

and passing through the point (0,9)

Construct a polynomial function that might have this graph.

The end behavior of the graph is determined by the coefficient and degree of highest degree term

Sketch the graph of these functions. What is the end behavior?

Leading term determines end behavior

1. For n even.

2. For n odd.

Rises to left

and falls to right

Rises to left

and rises to right

Falls to left

and falls to right

Falls to left

and rises to right

Given the polynomial

What is the degree of this polynomial?

1. End behavior.

2. x-intercepts. Solve f(x) = 0

Behavior at each intercept (even/odd)

b) If k > 1, graph flattens for larger values of k.

3. y-intercepts. Find f(0).

4. Symmetry: Odd/Even

5. Turning points: Graph changes between increasing/decreasing.